Constitutive Relations, Off Shell Duality Rotations and the Hypergeometric Form of Born-Infeld Theory

Abstract

We review equivalent formulations of nonlinear and higher derivatives theories of electromagnetism exhibiting electric-magnetic duality rotations symmetry. We study in particular on shell and off shell formulations of this symmetry, at the level of action functionals as well as of equations of motion. We prove the conjecture that the action functional leading to Born-Infeld nonlinear electromagnetism, that is duality rotation invariant off shell and that is known to be a root of an algebraic equation of fourth order, is a hypergeometric function. † Contribution to the proceedings of the conference BUDS2013, Breaking of supersymmetry and Ultraviolet Divergences in extended Supergravity,

Massive Born-Infeld and other dual pairs

“…This property of self-reproduction reflects, in fact, the invariance of the four-dimensional massless Maxwell system, or of generalizations thereof as the Born-Infeld system, under duality rotations, which translates into the condition [14,15,20,21] …”

“…In section 2 we describe massless and massive dualities in a general setting where the mass is induced by a Green-Schwarz coupling. In section 3 we describe the notion of self-dual actions [20,21], which is only available for D = 2p and p even, a particular case being the massive Born-Infeld system. In section 4 we turn to the peculiar self-dual gauge-field systems in odd dimensions, which are only possible for D = 2p + 1 and p odd.…”

“…(3.5). The first step involves the transition to a first-order for F and the introduction of two Lagrange multipliers λ and σ, as in [7,13,21]. The first multiplier eliminates the square root, while the second reduces the term depending on F F to a quadratic expression.…”

Constitutive relations and Schrödinger’s formulation of nonlinear electromagnetic theories

“…This is achieved doubling them (to G * = h[F, λ] and F * = k[G, λ]) and then constraining them via a symplectic matrix M. This matrix is well known in the study of duality rotations in linear electromagnetism coupled to scalar fields (see e.g. [35]). Here M will be in general dependent on the field strengths F, G and their derivatives, leading to nonlinear and higher derivatives electromagnetism.…”

“…Since the expansions for weak and slowly varying fields are expansions in adimensional variables (like for example λF F and λF F * , or, schematically and using a different coupling constant, λ∂F ∂F ) we will equivalently say that these expansions are in power series of the coupling constant(s) λ. 3 The factor 2 is due to the convention ∂Fρσ ∂Fµν = δ µ ρ δ ν σ adopted in [7] and in the review [35]. It will be used throughout the paper.…”